Multiplicative Zagreb indices and coindices of ... - Semantic Scholar

Opuscula Math. 36, no. 3 (2016), 287–299 http://dx.doi.org/10.7494/OpMath.2016.36.3.287

Opuscula Mathematica

MULTIPLICATIVE ZAGREB INDICES AND COINDICES OF SOME DERIVED GRAPHS Bommanahal Basavanagoud and Shreekant Patil Communicated by Dalibor Fronček Abstract. In this note, we obtain the expressions for multiplicative Zagreb indices and coindices of derived graphs such as a line graph, subdivision graph, vertex-semitotal graph, edge-semitotal graph, total graph and paraline graph. Keywords: multiplicative Zagreb indices and coindices, derived graphs. Mathematics Subject Classification: 05C07.

1. INTRODUCTION In this paper, we are concerned with simple graphs without isolated vertices. Let G be such a graph with vertex set V (G), |V (G)| = n, and edge set E(G), |E(G)| = m. As usual, n is the order and m the size of G. The degree of a vertex w ∈ V (G) is the number of vertices adjacent to w and is denoted by dG (w). A vertex w ∈ V (G) is said to be pendant if dG (w) = 1. The degree of an edge e = uv in G, denoted by dG (e), is defined by dG (e) = dG (u) + dG (v) − 2. We refer to [9] for unexplained terminology and notation. A graphical invariant is a number related to a graph, in other words, it is a fixed number under graph automorphisms. In chemical graph theory, these invariants are also called the topological indices. In 1984, Narumi and Katayama [11] considered the product index as Y N K(G) = dG (u) u∈V (G)

for representing the carbon skeleton of a saturated hydrocarbon, and named it as a simple topological index. Tomović and Gutman renamed this molecular structure c AGH University of Science and Technology Press, Krakow 2016

287

288

Bommanahal Basavanagoud and Shreekant Patil

descriptor as the Narumi-Katayama index [15]. In 2010, Todeshine et al. [13, 14] proposed the multiplicative variants of ordinary Zagreb indices, which are defined as follows: Y Y Y Y (G) = dG (u)2 = [N K(G)]2 and (G) = dG (u)dG (v). 1

2

u∈V (G)

uv∈E(G)

These two graph invariants are called first and second multiplicative Zagreb indices by Gutman [6]. And recently, Eliasi et al. [5] introduced a further multiplicative version of the first Zagreb index as Y∗ Y (G) = [dG (u) + dG (v)]. 1

uv∈E(G)

In [18] and [7] the authors called it a multiplicative sum Zagreb index and modified first multiplicative Zagreb index respectively. The second multiplicative Zagreb index for any graph G can also be written as [6] Y Y (G) = dG (u)dG (u) . 2

u∈V (G)

Xu et al. [19] defined the first and second multiplicative Zagreb coindices, respectively, as Y

1

(G) =

Y

[dG (u) + dG (v)]

uv6∈E(G)

and

Y

2

(G) =

Y

dG (u)dG (v).

uv6∈E(G)

The main properties of multiplicative Zagreb indices are summarized in [1, 4, 10, 12, 17, 18, 20]. We introduce the modified second multiplicative Zagreb index as Y∗ Y (G) = [dG (u) + dG (v)][dG (u)+dG (v)] . 2

uv∈E(G)

2. DERIVED GRAPHS In recent papers [2, 3, 8], the authors obtained the expressions for Zagreb indices Q Q Q∗ and coindices of derived graphs. This motivates us to find expressions for 1 , 2 , 1 and Q 2 of derived graphs. Let G be a graph with vertex set V (G) and edge set E(G). We are concerned with the following graphs derived from G ([8]): – line graph L = L(G); V (L) = E(G) and the two vertices of L are adjacent if the corresponding edges of G are incident with a common vertex; – subdivision graph S = S(G); V (S) = V (G) ∪ E(G) and the vertex of S corresponding to the edge uv of G is inserted in the edge uv of G;

289

Multiplicative Zagreb indices and coindices. . .

– vertex-semitotal graph T2 = T2 (G); V (T2 ) = V (G) ∪ E(G) and E(T2 ) = E(S) ∪ E(G); – edge-semitotal graph T1 = T1 (G); V (T1 ) = V (G)∪E(G) and E(T1 ) = E(S)∪E(L); – total graph T = T (G); V (T ) = V (G) ∪ E(G) and E(T ) = E(S) ∪ E(G) ∪ E(L); – paraline graph P L = P L(G) is the line graph of the subdivision graph. In Figure 1 self-explanatory examples of these derived graphs are depicted. b

b

G:

b b

b b b b r

r

r

b r r

L:

b

r r

r b

S: r

T2 : r

r

r r

b

b r

r b

r r b

r

T :

b r r

b b

r b r b

b b

r b

b

b

r b

r b

r

T1 :

b b

r

b

r

r

b b

PL :

b

b

b b

b b b

r

b b

b

Fig. 1. Various graphs derived from the graph G. The vertices of these derived graphs (except the paraline graph P L), corresponding to the vertices of the parent graph G, are indicated by circles. The vertices of these graphs corresponding to the edges of the parent graph G are indicated by squares

In [19], Kexiang Xu et al. obtained the expressions for graph G as Y

2

(G) =

Y

u∈V (G)

dG (u)n−1−dG (u)

and

Y

2

Q

2 (G)

Y (G) (G) = 2

of any connected

Y

1

! n−1 2

(G)

which are not satisfied for a complete graph. The following lemmas give the correct Q expressions for 2 (G).

290


Lemma 2.1. For a connected graph G 6= Kn , we have Y

2

Y

(G) =

dG (u)n−1−dG (u) .

u∈V (G)

Lemma 2.2. For a connected graph G 6= Kn , we have Y

2

Y (G) (G) = 2

Y

1

! n−1 2

(G)

.

Next we present the values of multiplicative Zagreb indices and coindices for several classes of graphs. Example 2.3. Let Pn be the path with n vertices. The pendant vertices have degree 1 and other vertices have degree 2. Hence, Q (i) Q1 (Pn ) = 4(n−2) , (ii) Q2 (Pn ) = 4(n−2) , ∗ (iii) 1 (Pn ) = 9 · 4(n−3) , n ≥ 3, Q (iv) 1 (Pn ) = 2 · 9(n−3) · 4(n−4)! , n ≥ 4, Q (v) Q2 (Pn ) = 2(n−2)(n−3) , n ≥ 3, ∗ (vi) 2 (Pn ) = 36 · 44(n−3) , n ≥ 3. Example 2.4. Consider the cycle Cn with n vertices. Since its every vertex is of degree 2, then Q (i) Q1 (Cn ) = 4n , (ii) Q2 (Cn ) = 4n , ∗ (iii) 1 (Cn ) = 4n , Q n(n−3) (iv) 1 (Cn ) = 4 2 , n ≥ 4, Q n(n−3) (v) Q2 (Cn ) = 4 2 , n ≥ 4, ∗ (vi) 2 (Cn ) = 44n .

Example 2.5. Let Kn be the complete graph on n vertices. All vertices of Kn have degree n − 1 and so Q (i) Q1 (Kn ) = (n − 1)2n , n ≥ 2, (ii) 2 (Kn ) = (n − 1)n(n−1) , n ≥ 2, Q∗ n(n−1) (iii) 1 (Kn ) = [2(n − 1)] 2 , n ≥ 2, Q (iv) 1 (Kn ) = 0, Q (v) 2 (Kn ) = 0, Q∗ 2 (vi) 2 (Kn ) = [2(n − 1)]n(n−1) , n ≥ 2. Example 2.6. Let Kr,s be the complete bipartite graph. Then Kr,s has r + s vertices and rs edges. Hence,

291


(i) (ii) (iii) (iv) (v) (vi)

Q 2s · s2r , Q1 (Kr,s ) = r rs Q∗2 (Kr,s ) = [rs] , rs (Kr,s ) = [r + s] , Q1 s(s−1) r(r−1) (K ) = [2r] 2 · [2s] 2 , r 6= 1 and s 6= 1, Q1 r,s s(s−1) · sr(r−1) , r 6= 1 and s 6= 1, Q2∗ (Kr,s ) = r rs(r+s) . 2 (Kr,s ) = [r + s]

Example 2.7. Let Wn be the wheel on n vertices. Its central vertex has degree n − 1 and its other vertices have degree 3. This implies Q (i) Q1 (Wn ) = (n − 1)2 · 32(n−1) , (ii) Q2 (Wn ) = [3(n − 1)](n−1) · 32(n−1) , ∗ (iii) 1 (Wn ) = [n + 2](n−1) · 6(n−1) , Q (n−1)(n−4) 2 , n ≥ 5, (iv) 1 (Wn ) = 6 Q (n−1)(n−4) 2 , n ≥ 5, (v) Q2 (Wn ) = 9 ∗ (vi) 2 (Wn ) = 66(n−1) · [n + 2](n−1)(n+2) . 3. RESULTS Theorem 3.1. Let G be a graph of order n and size m. Then Y Y (G). (S) = 4m 1

1

Proof. Note that S has n + m vertices. Y Y Y (S) = dS (u)2 = 1

u∈V (S)

dS (u)2

u∈V (S)∩V (G)

Y

dS (e)2 .

e∈V (S)∩E(G)

Note that for u ∈ V (S) ∩ V (G), dS (u) = dG (u) and for e ∈ V (S) ∩ E(G), dS (e) = 2. Y Y Y Y (G). (S) = dG (u)2 22 = 4m 1

u∈V (G)

e∈E(G)

1

Theorem 3.2. Let G be a graph of order n and size m. Then Y Y (S) = 4m (G). 2

Proof. Since S has n + m vertices, then Y Y Y (S) = dS (u)dS (u) = 2

u∈V (S)

2

dS (u)dS (u)

u∈V (S)∩V (G)

Y

dS (e)dS (e) .

e∈V (S)∩E(G)

Since for u ∈ V (S) ∩ V (G), dS (u) = dG (u) and for e ∈ V (S) ∩ E(G), dS (e) = 2. Y Y Y Y (S) = dG (u)dG (u) (G). 22 = 4m 2

u∈V (G)

e∈E(G)

2

292


Theorem 3.3. Let G be a graph of order n and size m. Then Y∗

Y

(S) =

1

[2 + dG (u)]dG (u) .

u∈V (G)

Proof. Since S has n + m vertices, then we have Y∗ 1

Y

(S) =

[dS (u) + dS (e)].

uv∈E(S)

Since for u ∈ V (S) ∩ V (G), dS (u) = dG (u) and for e ∈ V (S) ∩ E(G), dS (e) = 2. Y∗

Y

(S) =

1

[2 + dG (u)]dG (u) .

u∈V (G)

Corollary 3.4. Let G be a connected graph of order n and size m. Then Y

2

2m

(S) =

2

+nm−3m

Q

Proof. From Lemma 2.2 we have Y

2

(S) =

[

Q

[

Q

1 (G)]

n+m−1 2

2 (G)

(S)] 1Q

n+m−1 2

2 (S)

.

.

From Theorems 3.1 and 3.2 we get the result.

Theorem 3.5. Let G be a graph of order n and size m. Then Y

1

(T2 ) = 4n+m

Proof. Note that T2 has n + m vertices. Y

1

Y

(T2 ) =

dT2 (u)2 =

u∈V (T2 )

Y

Y

1

(G).

dT2 (u)2

u∈V (T2 )∩V (G)

Y

dT2 (e)2 .

e∈V (T2 )∩E(G)

Note that for u ∈ V (T2 )∩V (G), dT2 (u) = 2dG (u) and for e ∈ V (T2 )∩E(G), dT2 (e) = 2. Y

1

(T2 ) =

Y

[2dG (u)]2

u∈V (G)

Y

22 = 4n+m

e∈E(G)

Y


2

(T2 ) = 64m

Y

1

(G)

Y

2

(G).

1

(G).

293


Proof. Since T2 has n + m vertices and 3m edges, then we have Y

2

Y

(T2 ) =

dT2 (u)dT2 (v)

uv∈E(T2 )

Y

=

Y

dT2 (u)dT2 (v)

uv∈E(T2 )∩E(G)

dT2 (u)dT2 (e).

ue∈E(T2 )\E(G)

Since for u ∈ V (T2 ) ∩ V (G), dT2 (u) = 2dG (u) and for e ∈ V (T2 ) ∩ E(G), dT2 (e) = 2. Y

2

Y

(T2 ) =

2dG (u)2dG (v) Y

dG (u)dG (v)42m

uv∈E(G)

= 64

m

Y

1

(2)2dG (u)

ue∈E(T2 )\E(G)

uv∈E(G)

= 4m

Y

(G)

Y

Y

dG (u)2

u∈V (G) 2

(G).

Theorem 3.7. Let G be a graph of order n and size m. Then Y∗ 1

(T2 ) = 8m

Y∗ 1

(G)

Y

[1 + dG (u)]dG (u) .

u∈V (G)

Proof. Since T2 has n + m vertices and 3m edges, then we have Y∗ 1

(T2 ) =

Y

[dT2 (u) + dT2 (v)]

uv∈E(T2 )

=

Y

Y

[dT2 (u) + dT2 (v)]

uv∈E(T2 )∩E(G)

[dT2 (u) + dT2 (e)].

ue∈E(T2 )\E(G)

Since for u ∈ V (T2 ) ∩ V (G), dT2 (u) = 2dG (u) and for e ∈ V (T2 ) ∩ E(G), dT2 (e) = 2. Y∗ 1

Y

(T2 ) =

[2dG (u) + 2dG (v)]

=8

m

Y

[dG (u) + dG (v)]22m

uv∈E(G) Y∗ 1

[2 + 2dG (u)]

ue∈E(T2 )\E(G)

uv∈E(G)

= 2m

Y

(G)

Y

Y

[1 + dG (u)]dG (u)

u∈V (G) dG (u)

[1 + dG (u)]

.

u∈V (G)

Corollary 3.8. Let G be a connected graph of order n and size m. Then Y

2

(T2 ) =

2(n+m)

2

Q n+m−3 [ 1 (G)] 2 Q . 2 (G)

−n−7m

294



2

[

(T2 ) =

Q

(T2 )] 1Q

n+m−1 2

2 (T2 )

.



1

(T1 ) =

Y

1

Proof. Note that T1 has n + m vertices. Y

1

Y

(T1 ) =

h Y∗ i2 (G) (G) . 1

dT1 (u)2

u∈V (T1 )

Y

=

Y

dT1 (u)2

u∈V (T1 )∩V (G)

dT1 (ei )2 .

ei ∈V (T1 )∩E(G)

Note that for u ∈ V (T1 ) ∩ V (G), dT1 (u) = dG (u) and for ei ∈ V (T1 ) ∩ E(G), dT1 (ei ) = dG (ui ) + dG (vi ). Y

1

Y

(T1 ) =

u∈V (G)

=

Y

1

Y

dG (u)2

[dG (ui ) + dG (vi )]2

ui vi ∈E(G)

h Y∗ i2 (G) (G) . 1


2

(T1 ) =

Y

2

(G)

Y∗ 2

(G).

Proof. Since T1 has n + m vertices, then we have Y

2

Y

(T1 ) =

dT1 (u)dT1 (u)

u∈V (T1 )

Y

=

Y

dT1 (u)dT1 (u)

u∈V (T1 )∩V (G)

[dT1 (ei )][dT1 (ei )] .

ei ∈V (T1 )∩E(G)

Since for u ∈ V (T1 ) ∩ V (G), dT1 (u) = dG (u) and for ei ∈ V (T1 ) ∩ E(G), dT1 (ei ) = dG (ui ) + dG (vi ). Y

2

(T1 ) =

Y

dG (u)dG (u)

u∈V (G)

=

Y

2

(G)

Y∗ 2

Y

[dG (ui ) + dG (vi )][dG (ui )+dG (vi )]

ui vi ∈E(G)

(G).

295


Corollary 3.11. Let G be a connected graph of order n and size m. Then Q n+m−1 Q∗ Y [ 1 (G)] 2 [ 1 (G)]n+m−1 Q Q∗ (T1 ) = . 2 2 (G) 2 (G) Proof. From Lemma 2.2 we have Y

2

(T1 ) =

[

Q

(T1 )] 1Q

n+m−1 2

2 (T1 )

.

From Theorems 3.9 and 3.10 we get the result. Q In [16], an incorrect expression Q for 1 (T ) was established. The following theorem gives the correct expression for 1 (T ).

Theorem 3.12. Let G be a graph of order n and size m. Then h Y∗ i2 Y Y (G) (T ) = 4n (G) . 1

1

1

Proof. Note that T has n + m vertices. Y Y Y (T ) = dT (u)2 = 1

u∈V (T )

dT (u)2

u∈V (T )∩V (G)

Y

dT (ei )2 .

ei ∈V (T )∩E(G)

Note that for u ∈ V (T ) ∩ V (G), dT (u) = 2dG (u) and for ei ∈ V (T ) ∩ E(G), dT (ei ) = dG (ui ) + dG (vi ). h Y∗ i2 Y Y Y Y (T ) = [2dG (u)]2 [dG (ui ) + dG (vi )]2 = 4n (G) (G) . 1

u∈V (G)

1

ui vi ∈E(G)

1

Theorem 3.13. Let G be a graph of order n and size m. Then hY i2 Y Y∗ (T ) = 16m (G) (G) . 2

2

2

Proof. Since T has n + m vertices, then we have Y Y Y (T ) = dT (u)dT (u) = dT (u)dT (u) 2

u∈V (T )

u∈V (T )∩V (G)

Y

dT (ei )dT (ei ) .

ei ∈V (T )∩E(G)

Note that for u ∈ V (T ) ∩ V (G), dT (u) = 2dG (u) and for ei ∈ V (T ) ∩ E(G), dT (ei ) = dG (ui ) + dG (vi ). Y Y Y (T ) = [2dG (u)][2dG (u)] [dG (ui ) + dG (vi )][dG (ui )+dG (vi )] 2

u∈V (G)

=

Y

ui vi ∈E(G)

2[2dG (u)] [dG (u)]2dG (u)

u∈V (G)

= 16m

Y∗ 2

i2 hY (G) . (G) 2

Y∗ 2

(G)

296


Corollary 3.14. Let G be a connected graph of order n and size m. Then Q n+m−1 Q∗ Y 2n(n+m−1)−4m [ 1 (G)] 2 [ 1 (G)]n+m−1 Q∗ Q . (T ) = 2 2 2 (G)[ 2 (G)] Proof. From Lemma 2.2 we have Y

2

(T ) =

[

Q

(T )] 1Q

n+m−1 2

2 (T )

.

From Theorems 3.12 and 3.13 we get the result. Theorem 3.15. Let G be a graph of order n and size m. Then i2 hY Y (G) . (P L) = 2

1

Proof. Note that paraline graph P L has 2m vertices, and dG (u) of its vertices have the same degree as the vertex u of the graph G. i2 hY Y Y Y (P L) = dP L (u)2 = (G) . dG (u)[2dG (u)] = 1

u∈V (P L)

2

u∈V (G)

Theorem 3.16. Let G be a graph of order n and size m. Then Y Y 2 (P L) = [dG (u)][dG (u)] . 2

u∈V (G)

Proof. Since P L has 2m vertices, then we have Y Y (P L) = dP L (u)dP L (v) 2

uv∈E(P L)

=

Y

2

(G)

Y

[dG (u)][dG (u)(dG (u)−1)] =

Y

u∈V (G)

Theorem 3.17. Let G be a graph of order n and size m. Then Y∗ Y∗ Y dG (u)(dG (u)−1) 2 . (P L) = (G) [2dG (u)] 1

2

[dG (u)][dG (u)] .

1

u∈V (G)

Proof. Since P L has 2m vertices, then we have Y∗ Y Y∗ Y dG (u)(dG (u)−1) 2 (P L) = [dP L (u) + dP L (v)] = (G) [2dG (u)] . 1

1

uv∈E(P L)

u∈V (G)

Corollary 3.18. Let G be a connected graph of order n and size m. Then Q Y [ 2 (G)]2m−1 (P L) = Q 2 [dG (u)][dG (u)]2 u∈V (G)

297



2

(P L) =

[

Q

(P L)] 1Q

2m−1 2

2 (P L)

.


One can easily obtain the expressions for multiplicative Zagreb indices and coindices of line graph L of graph G by considering edge degrees of G. Theorem 3.19. Let G be a graph of order n and size m. Then Q Q dG (e)2 , (i) 1 (L) = e∈E(G) Q Q (ii) 2 (L) = dG (ei )dG (ej ), ei ∼ej Q∗ Q (iii) 1 (L) = [dG (ei ) + dG (ej )], ei ∼ej Q Q [dG (ei ) + dG (ej )], (iv) 1 (L) = ei ≁ej Q Q (v) 2 (L) = dG (ei )dG (ej ), ei ≁ej

where ei ∼ ej (resp. ei ≁ ej ) means that the edges ei and ej are adjacent (resp. not adjacent) in G. Q∗ Q∗ It remains a task for the future to find the expressions for 1 (T1 ) and 1 (T ). T Q In [19], the total multiplicative sum Zagreb index (G) of a graph G is defined as T Y (G) =

Y

[dG (u) + dG (v)].

u,v∈V (G)

Lemma 3.20 ([19]). For a connected graph G, we have

T Q Q (G) = (G). (G) 1 1

Q∗

Q By Lemma 3.20, one can find the expression for 1 of derived graphs. But obtainT Q ing the expression for of derived graphs is a difficult task.

Acknowledgments The authors would like to thank the referee for his/her careful reading and useful suggestions which led us to improve the paper. This research is supported by UGC-MRP, New Delhi, India: F.No.41-784/2012 dated: 17-07-2012. This research is supported by UGC-UPE (Non-NET)-Fellowship, K. U. Dharwad, No. KU/Sch/UGC-UPE/2014-15/897, dated: 24 Nov 2014.

298


REFERENCES [1] M. Azari, A. Iranmanesh, Some inequalities for the multiplicative sum Zagreb index of graph operations, Journal of Mathematical Inequalities 9 (2015) 3, 727–738. [2] B. Basavanagoud, I. Gutman, V.R. Desai, Zagreb indices of generalized transformation graphs and their complements, Kragujevac J. Sci. 37 (2015), 99–112. [3] B. Basavanagoud, I. Gutman, C.S. Gali, On second Zagreb index and coindex of some derived graphs, Kragujevac J. Sci. 37 (2015), 113–121. [4] K.C. Das, A. Yurttas, M. Togan, A.S. Cevik, I.N. Cangul, The multiplicative Zagreb indices of graph operations, Journal of Inequality and Applications (2013), 2013:90. [5] M. Eliasi, A. Iranmanesh, I. Gutman, Multiplicative versions of first Zagreb index, MATCH Commun. Math. Comput. Chem. 68 (2012), 217–230. [6] I. Gutman, Multiplicative Zagreb indices of trees, Bull. Internat. Math. Virt. Inst. 1 (2011), 13–19. [7] I. Gutman, Degree-based topological indices, Croat. Chem. Acta 86 (2013), 351–361. [8] I. Gutman, B. Furtula, Ž. Kovijanić Vukićević, G. Popivoda, Zagreb indices and coindices, MATCH Commun. Math. Comput. Chem. 74 (2015), 5–16. [9] F. Harary, Graph Theory, Addison-Wesley, Reading, Mass 1969. [10] J. Liu, Q. Zhang, Sharp upper bounds for multiplicative Zagreb indices, MATCH Commun. Math. Comput. Chem. 68 (2012), 231–240. [11] H. Narumi, M. Katayama, Simple topological index. A newly devised index characterizing the topological nature of structural isomers of saturated hydrocarbons, Mem. Fac. Engin. Hokkaido Univ. 16 (1984), 209–214. [12] T. Réti, I. Gutman, Relations between ordinary and multiplicative Zagreb indices, Bull. Internat. Math. Virt. Inst. 2 (2012), 133–140. [13] R. Todeschini, V. Consonni, New local vertex invariants and molecular descriptors based on functions of the vertex degrees, MATCH Commun. Math. Comput. Chem. 64 (2010), 359–372. [14] R. Todeschini, D. Ballabio, V. Consonni, Novel molecular descriptors based on functions of new vertex degrees, [in:] I. Gutman, B. Furtula (eds), Novel molecular structure descriptors – Theory and applications I, Univ. Kragujevac, 2010, 73–100. [15] Ž. Tomovič, I. Gutman, Narumi-Katayama index of phenylenes, J. Serb. Chem. Sco. 66 (2001) 4, 243–247. [16] H. Wang, H. Bao, A note on multiplicative sum Zagreb index, South Asian J. Math. 2 (2012) 6, 578–583. [17] S. Wang, B. Wei, Multiplicative Zagreb indices of k-tree, Discrete Applied Math. 180 (2015), 168–175. [18] K. Xu, K.C. Das, Trees, unicyclic and bicyclic graphs extremal with respect to multiplicative sum Zagreb index, MATCH Commun. Math. Comput. Chem. 68 (2012), 257–272.


299

[19] K. Xu, K.C. Das, K. Tang, On the multiplicative Zagreb coindex of graphs, Opuscula Math. 33 (2013) 1, 191–204. [20] K. Xu, H. Hua, A unified approach to extremal multiplicative Zagreb indices for trees, unicyclic and bicyclic graphs, MATCH Commun. Math. Comput. Chem. 68 (2012), 241–256.

Bommanahal Basavanagoud [email protected] Karnatak University Department of Mathematics Dharwad – 580 003, Karnataka, India

Shreekant Patil [email protected] Karnatak University Department of Mathematics Dharwad – 580 003, Karnataka, India

Received: May 2, 2015. Revised: September 24, 2015. Accepted: October 13, 2015.